The screenshot below suggests thatan ITM put option's price can't overstep its strike price? Why or why not?
Under the assumption that the underlying cannot have a negative value, then the value of a put option cannot be greater than the strike.
The reason behind this doesn't require maths, it's fairly simple: the lowest possible value of the underlying is zero. At that price, you make the maximum possible profit, of K. The value of the option is the probability weighted average of the payoffs - we have just explained that the maximum payoff is K, so there is no possible probability distribution that can average more than that. Therefore the maximum possible value is the maximum payoff, which is K.
If we remove that assumption that the underlying can trade negative, then the above reasoning goes away.
This happened when the possibility of rates going negative first became apparent, and suddenly zero strike swaptions became a real thing.
It happened again earlier this year when oil futures traded down to negative \$40/bbl. For a couple of months after that several options with strikes near zero (ie the 50c, \$1, \$1.50) traded at prices above their strikes. The exchanges (nymex and ice) actually listed negative strike options, though the volume that traded on them was extremely small, and market makers didn't go anywhere near them.
EDIT: and to complete the answer, i guess i should include @Quantuple's comment above - if you have negative interest rates, such that \$1 today is worth less than \$1 in the future, then an option of any strike can ahve a value higher than the strike, if your time to expiry is large enough.